Model the cities as nodes in an undirected graph and count the number of connected components. Use either DFS/BFS to traverse each component or Union-Find to merge connected cities, then count the distinct sets.
Pro tip: Clarify whether the graph is undirected (adjacency matrix is symmetric) and mention that you can optimize space by using a 1D visited array instead of a full matrix copy. Also, briefly discuss trade-offs between DFS and Union-Find for large n.
Confirm that the adjacency matrix represents an undirected graph where matrix[i][j] = 1 means city i and city j are directly connected. The goal is to count the number of connected components (provinces).
Decide between DFS/BFS and Union-Find. DFS/BFS is straightforward for counting components; Union-Find is efficient for dynamic connectivity but overkill here. Mention both and pick one based on constraints.
For DFS/BFS: iterate over all cities, and for each unvisited city, increment the component count and traverse all reachable cities, marking them visited. For Union-Find: initialize each city as its own parent, union connected cities, then count distinct roots.
State time and space complexity: DFS/BFS is O(n^2) time and O(n) space; Union-Find with path compression and union by rank is O(n^2 α(n)) time and O(n) space.
Walk through a small example (e.g., n=3 with matrix [[1,1,0],[1,1,0],[0,0,1]]) to verify the count is 2. Discuss edge cases like n=1 or fully connected graph.
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